Theorem int0 4855

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4408	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3413	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4848	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 234	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 267	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2755	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409  ∅c0 4227  ∩ cint 4841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-dif 3863  df-nul 4228  df-int 4842

This theorem is referenced by:  unissint  4865  uniintsn  4880  rint0  4883  intex  5211  intnex  5212  oev2  8164  fiint  8841  elfi2  8924  fi0  8930  cardmin2  9474  00lsp  19835  cmpfi  22122  ptbasfi  22295  fbssint  22552  fclscmp  22744  zarcmplem  31365  rankeq1o  34057  bj-0int  34831  heibor1lem  35562